A short note on compact embeddings of reproducing kernel Hilbert spaces in $L^2$ for infinite-variate function approximation

This note consists of two largely independent parts. In the first part we give conditions on the kernel $k: \Omega \times \Omega \rightarrow \mathbb{R}$ of a reproducing kernel Hilbert space $H$ continuously embedded via the identity mapping into $L^2(\Omega, \mu),$ which are equivalent to the fact that $H$ is even compactly embedded into $L^2(\Omega, \mu).$ In the second part we consider a scenario from infinite-variate $L^2$-approximation. Suppose that the embedding of a reproducing kernel Hilbert space of univariate functions with reproducing kernel $1+k$ into $L^2(\Omega, \mu)$ is compact. We provide a simple criterion for checking compactness of the embedding of a reproducing kernel Hilbert space with the kernel given by $$\sum_{u \in \mathcal{U}} \gamma_u \bigotimes_{j \in u}k,$$ where $\mathcal{U} = \{u \subset \mathbb{N}: |u| < \infty\},$ and $(\gamma_u)_{u \in \mathcal{U}}$ is a sequence of non-negative numbers, into an appropriate $L^2$ space.